Effect of surface energy on the non-linear postbuckling behavior of nanoplates
Introduction
Owing to the requirement of miniaturization of electromechanical systems, the physical and mechanical behaviors of nanostructures must be exactly understood to ensure their reliable applications. The surface to volume ratio of nanoscale materials is large and the influence of surface effect on the mechanical properties of such materials is significant. For examples, Atomistic simulation has shown that the elastic modulus of nanoplates can be larger or smaller than their bulk counterparts due to the effect of surface elasticity [1]. Liang and Jiang [2], Miller and Shenoy [3], and McDowell et al. [4] studied the mechanical behaviors of various nanostructures and found that their elastic constants are size-dependent. The study of mechanical behavior of material with surface effects has received increasing interests in more recent researches [5], [6], [7].
On the other hand, plate-like nanostructures have been widely used in micro-electromechanical systems (MEMS) and nano-electromechanical systems (NEMS) [8]. For these applications, nanoplates can be considered as two dimensional ultrathin films which can be produced by special methods [9], [10], [11], [12]. Naturally, it is important to understand the mechanical properties of nanoscale plates. Lu et al. [13] proposed a general thin plate theory including surface residual stress and surface elasticity, which can be used for size-dependent static and dynamic analysis of plate-like thin film structures. Wang and Zhao [14] investigated the size-dependent self-buckling of nanoplates with surface effect. Aksencer and Aydogdu [15] studied the buckling of nanoplates by using non-local elasticity theory.
In most situations, nanostructures are subjected to large deformations. The deflections of a nanoplate can be of the order of its thickness. In this situation the infinitesimal deformation model will no longer be valid. It is necessary to apply the geometrically non-linear model for the analysis of such small nanostructures. The geometrically non-linear model was used to study the surface effects on the postbuckling of nanowires [16], [17], [18], [19]. However, geometrically non-linear analysis for the influence of surface effect on the mechanical properties of two dimensional nanoplates is rare. To our best knowledge, only Lim and He [20] used the Von Karman type non-linear model to study the static mechanical properties of ultra-thin isotropic films with surface effect. Since plate-like nanostructures commonly sustain large deformations, it is necessary and significant to investigate the influence of surface effect on the postbuckling behavior of nanoplates. However, no study was conducted for the postbuckling behavior of nanoplates with surface effect.
In this paper, we studied the surface effect on the postbuckling behavior of nanoplates by using Von Karman type non-linear model. We derived the governing equations and boundary conditions and obtained approximate postbuckling loads by using Galerkin’s method for Kirchhoff type plate and Mindlin type plate. The surface elasticity, residual surface stress and geometrically non-linear strain were considered simultaneously. The results indicate that the postbuckling load of the plates is size-dependent and can be enhanced significantly when the thickness of the plate decreases. Similar to the traditional continuum mechanics prediction, the transverse shear deformation also reduces the postbuckling load of the nanoplates.
Section snippets
Formulation
The surface stress constitutive relations can be derived based on the energy balance theory of material surfaces [3], [21]:where γ,ταβ and εαβ are, respectively, the surface energy density, the surface stresses and surface strains. According to Gurtin and Murdoch model [22] the constitutive and equilibrium relations of surface layer S± (assuming the surface layer S± have the same properties) can be expressed as
Solution by using Galerkin’s method
As usual a force function is defined byand a function L is introduced according to
Using Eqs. (15), (16) and , we can obtain the following compatibility equationwhere E*=E+2Es/h.
The equilibrium equations can be obtained asfor the Kirchhoff plate, andfor the Mindlin plate.
Now consider that edge conditions of a quadratic plate whose length is a
Approximate closed-form solutions
It has been argued that the results by using one-term approximation of Galerkin’s method are in reasonable agreement with those previously obtained [24], [25], [26]. Therefore, we take one-term approximation of Galerkin’s method to solve the postbuckling loads of simply supported nanoplates. In this situation, the approximate solutions for the postbuckling loads of Kirchhoff nanoplates and Mindlin nanoplates can be obtained in closed-form.
Numerical results and discussions
Numerical results are given for a plate with the following material properties, E=76 Gpa, υ=0.3, Es=1.22N/m, τ0=0.89N/m [27]. The shear correction factor is taken as k2=π2/12. The plate is subjected to a compression force Nx=N on its edges x=a and x=b. Relationship between the postbuckling load and the maximum net deflection f for a square plate (λ=1) are shown in Fig. 1, Fig. 2 for Kirchhoff plates (a/h=15), and in Fig. 3, Fig. 4 for Mindlin plates (a/h=8). Comparisons of the results from the
Conclusion
A large-deflection model has been developed to study the postbuckling behavior of nanoplates. The present model incorporates the Von Karman’s geometrically non-linear strain and the surface effect. The governing equations and boundary conditions of both Kirchhoff plates and the Mindlin plates were obtained by using the principle of minimum potential energy. The relationship between the normalized postbuckling load and the maximum deflection of the nanoplates was established by using Galerkin’s
Acknowledgment
The authors are grateful to the National Science Foundation of China (project no. 11172081) and Guangdong Natural Science Foundation, China (project no. S2012010010778) for supporting their research.
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